The cutoff phenomenon for ergodic Markov processes
Guan-Yu Chen, Department of Applied Mathematics, National Chiao Tung University
Laurent Saloff-Coste, Department of Mathematics, Cornell University
Abstract
We consider the cutoff phenomenon in the context of families
of ergodic Markov transition functions.
This includes classical examples such as families
of ergodic finite Markov chains and Brownian motion on families
of compact Riemannian manifolds. We give criteria for the existence
of a cutoff when convergence is measured in Lp-norm with 1<p<∞.
This allows us to prove the existence of a cutoff in cases where
the cutoff time is not explicitly known. In the reversible case, for
1<p<∞, we show that a necessary and sufficient condition for
the existence of a max-Lp cutoff is that the product of the spectral gap
by the max-Lp mixing time tends to infinity. This type of condition
was suggested by Yuval Peres. Illustrative examples are
discussed.
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